The increasing need for higher data rates is continuously pushing the carrier frequencies of communication systems higher. This trend is complemented by the scaling down of device and interconnect dimensions that enable enhanced functionality on a single chip. The flip side of this continuing development is the increasing susceptibility of circuits to process variations, which produce lower yields.
Thus, as technology continues to shrink the size of electronic devices, the need to model the effect of process variations on circuit performance is becoming increasingly important. Process variations affect the performance of both active and passive components of many electronic devices or systems, which makes it necessary to include distributed interconnect structures for capturing the effect of process variations on overall circuit performance. Accordingly, it would be desirable to combine the process variations of circuits with the distributed interconnect-based passive components in a model, in order to capture the statistical behavior of overall device performance parameters, enabling the calculation of Probability Density Functions (PDFs) for key performance metrics of electronic devices and systems. As those of ordinary skill in statistical circuit analysis will appreciate, the PDF of a random variable is a measure of the probability of the random variable falling within a unit interval, at a specific value of the variable. The PDF for a parameter is often presented graphically and can be a discrete or a continuous function of the random variable. In the case of a discrete sample space, the PDF is called the probability mass function and is equal to the probability that the random variable takes on a specific value from the sample space.
The calculation of PDFs for one or more statistical variables during the development of an electronic device is important because a circuit designer can employ a PDF to evaluate the statistical behavior of a variable in regard to a proposed design and then modify the design as necessary to ensure that the yield of devices that meet specifications is at an acceptable level. Too often, this type of information is only developed after a design is entering the manufacturing stage, so that longer development times and substantial costs must then be incurred to modify the design, based on empirical results provided by initial production testing. Clearly, if an efficient technique were available to statistically evaluate a design in regard to key performance metrics, by calculating the PDF of one or more statistical variables before the design phase is concluded, products should be able to reach the marketplace much sooner, at a lower cost, and at a higher profitability to the manufacturer.
One technique for dealing with such problems is to use a combination of analytical techniques and response surface methodology to predict the statistical behavior of performance measures from the distributions of lower level process parameters. In another approach, a response surface technique has been used with a Technology CAD approach to determine the spread of circuit performance measures. In a recent work, the issues pertaining to asymptotic evaluation of the PDF of a random variable that can be expressed as a 2nd order response surface have been addressed.
However, conventional approaches to modeling statistical behavior are generally limited to circuit variables and have not attempted to combine circuit variables and electromagnetic (EM) variables of an electronic device to achieve a more accurate analysis. Modem Radio Frequency (RF) circuits operating at very high frequencies, e.g., at 10 GHz and above, such as on-chip inductors, need accurate modeling of passive components. While the characterization of distributed interconnect structures can be accomplished using field solvers, lumped elements and active components require the use of SPICE-like circuit simulators (the acronym “SPICE” refers to the well-known Simulation Program with Integrated Circuit Emphasis software).
Compared to the literature on statistical variability in circuits, little work has been undertaken in the area of statistical study on the performance of EM structures although this is of equal importance in present day design manufacturability analysis. The impact of variability on board level signal integrity using time domain reflectometry (TDR) measurements, field solvers and resistance, inductance, conductance, and capacitance (RLGC) transmission line models in HSPICE™ has been addressed. In another work, a commercial field simulator and linear regression tool have been used to perform statistical analysis of filters on liquid crystal polymers (LCP) substrates. In view of the large dimensions compared to on-chip distributed structures, the percentage standard deviations compared to mean values is small, allowing linear approximations for the objective functions in terms of the varying parameters, and thereby producing Gaussian PDFs. It has been found that for large standard deviations of parameters contributing to variability of on-chip passive components, quadratic response surfaces are more appropriate. There is a noticeable deviation of the quadratic response surfaces from the Gaussian profile for the desired objective functions. This observation is true for both circuit and EM variability.
With increasing frequency of operation of chips, passive components, such as on-chip spiral inductors, need to be modeled using field solvers in order to accurately capture all of the EM effects. Since process variations affect both circuits and EM structures or components, there is a clear requirement for an automated way of combining the two analysis tools to predict the impact of process variations on the overall circuit performance. One approach to accomplish this goal might be to also include parametric circuit models for the EM components in a SPICE-like simulator when constructing the response surface. There are two impediments associated with this approach. First, using circuit models for EM objects requires the complex step of generating parameterized, passive, and accurate RLC models at high frequencies. Second, since process variations are becoming greater, even for EM objects, the PDFs of the parameters defining the passives will be non-Gaussian and correlated in addition to being numerically computed. It is very difficult to generate samples from such distributions.
Accordingly, there are two critical issues for high frequency RF circuits in emerging technologies that need to be addressed. As noted above, particularly in regard to RF electronic devices, a method is needed to implement statistical analysis of combined circuit-EM systems. In addition, a method is needed that can address the challenge of non-Gaussian, correlated parameters representing the electromagnetically modeled passive components, as well as Gaussian and uncorrelated parameters.